Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1195the connection 1–form A, and get a Lie algebra element. As we have seen,it is precisely this Lie algebra element which transports vectors in E alongthis small distance: we have to multiply these vectors by 1 + A. This is alinear approximation to the finite transformation e A . So if we transport avector along the entire closed curve γ, it will return multiplied by a groupelementg = limδt→0[exp (A ( ˙γ(0)) δt) exp (A ( ˙γ(δt)) δt) exp (A ( ˙γ(2δt)) δt) · · · ] .Now it is tempting to add all the exponents and write their sum in thelimit as an integral, but this is not quite allowed since the different groupelements may not commute, so e X e Y ≠ e X+Y . Therefore, one uses thefollowing notation,g = P exp∫A,γwhere P stands for path ordering, while the element g is called the holonomyof A around the closed curve γ. An interesting gauge and metricindependent object turns out to be the trace of this group element. Thistrace is called the Wilson loop W γ (A), given by∫W γ (A) = Tr(P ) exp A.The topological invariants we are interested in are now the correlation functionsof such Wilson loops in Chern–Simons theory. Since these correlationfunctions are independent of the parametrization of M, we can equivalentlysay that they will be independent of the precise location of the loop γ; wehave in fact constructed a topological invariant of the embedding of γ insideM. This embedding takes the shape of a knot, so the invariants wehave constructed are knot invariants. One can show that the invariants areactually polynomials in the variable y = exp 2πi/(k + 2), where k is theinteger ‘coupling constant’ of the Chern–Simons theory.The above construction is due to E. Witten, and was carried out in[Witten (1988a)]. Before Witten’s work, several polynomial invariants ofknots were known, one of the simplest ones being the so–called Jones polynomial.It can be shown that many of these polynomials arise as specialcases of the above construction, where one takes a certain structure groupG, SU(2) for the Jones polynomial, and a certain vector bundle (representation)E, the fundamental representation for the Jones polynomial. Thatis, using this ‘trivial’ topological field theory, Witten was able to reproduceγ